This invention relates to density measurement using Compton scattering of X-rays for determining density at a point within an object without exposing the entire object to radiations. This invention also pertains to formulating an algorithm for solving density-measurement equations.
In nondestructive evaluation, it is often needed to know the density at a point, or points, in a region of interest within an object. Point-density measurement is useful, for instance, to detect a flaw in an isolated over-stressed region of a component. In another example, when a suspect material is identified by radiography, point-by-point imaging can be used to determine the density of the suspect material without having to generate a point-by-point density image of the entire object. This can be useful, for example, in the detection of explosives and other contraband materials in passenger luggage. In medical applications, point-by-point imaging can be useful in follow-up examination to determine, for instance, whether treatment was effective in destroying an isolated tumor.
Measuring density at an isolated point within an object using a X-ray beam requires the beam to reach the point of interest, to pass through the point of interest and to reach a detector. In its way to the point of interest, the beam is modified as it transverses other points, unless the point of interest is at the surface of the object. The beam is modified again by the point of interest. The beam is further affected by other points as it travels out of the object toward the detector. For this reason, basically, conventional transmission radiographic imaging is not suited for obtaining the density at a point within an object, since radiography provides an integrated line density along the path of the radiation beam penetrating the object. To determine the density at a point, many multiple radiation exposures at different angles or different directions must be effected, with subsequent numerical image reconstruction. This process is often referred to as computed tomography. Such a complete imaging process is tedious and expensive. It involves numerous consecutive measurements using many measuring devices and complex reconstruction algorithms to generate the image.
Examples of related prior art using radiographic imaging processes are described in the following patent documents:
U.S. Pat. No. 3,809,904 issued on May 7, 1974 to Clarke et al.;
U.S. Pat. No. 4,123,654 issued on Oct. 31, 1978 to Reiss et al.;
U.S. Pat. No. 4,228,351 issued on Oct. 14, 1980 to Snow et al.;
U.S. Pat. No. 4,768,214 issued on Aug. 30, 1988 to P. J. Bjorkholm;
U.S. Pat. No. 4,850,002 issued on Jul. 18, 1989 to Harding et al.;
U.S. Pat. No. 4,887,285 issued on Dec. 12, 1989 to Harding et al.;
U.S. Pat. No. 5,247,560 issued on Sept. 21, 1993 to Hosokama et al.;
U.S. Pat. No. 5,247,561 issued on Sept. 21, 1993 to A. F. Kotowski;
U.S. Pat. No. 5,696,806 issued on Dec. 9, 1997 to Grodzins et al.;
CA 1,101,133 issued on May 12,1981 to G. Harding;
CA 1,135,878 issued on Nov. 16, 1982 to Jatteau et al.;
CA 1,157,968 issued on Nov. 29, 1983 to Harding et al.
In the methods of the prior art, the attenuation of the radiation along the path of the X-ray beam is in most cases estimated, extrapolated from previous measurements or considered as a constant. It is believed that these estimations and extrapolations could lead to measurement inaccuracies, and for this reason, basically, it is believed that the prior art methods have only been used with limited degrees of success. As such, it may be appreciated that there continues to be a need for a method to determine with precision the density at a point, or points in a region of interest within an object, without performing a complete imaging of the object.
Before describing the present invention, however, it is deemed that certain general information should be reminded in order to afford a clearer understanding of the following specification. In particular, a general knowledge of the Compton scattering principle applicable to a X-ray beam is believed essential to facilitate the understanding of the present invention.
Compton scattering is the incoherent collision between photons and the free electrons of the atoms and it dominates all other photon interactions. Since Compton scattering is an interaction with the electrons of the atom, its probability of interaction depends on the density of the medium. Therefore, Compton scattering principle is available for non-destructive measurement of density.
In order to demonstrate how Compton scattering principle can be used to measure the density at a point within an object, reference is firstly made to FIG. 1 where a source of X-ray having an energy E is placed at point Ps and is directed at a small voxel V located at point Pv within an object O. A detector is placed at point Pd to determine the electron density of the voxel xe2x80x98Vxe2x80x99. In order for the detector at point Pd to monitor the scattered radiation Exe2x80x2 from point Pv it must be collimated so that it focuses along the direction Pv-Pd. The unique relationship between the scattered photon energy Exe2x80x2, and the scattering angle xcex8 is expressed as follows:
Exe2x80x2=E/(1+((E/moc2)(1xe2x88x92cos xcex8)))xe2x80x83xe2x80x83(1) 
where E is the initial energy of the incident photon, and moc2 is the rest mass of the electron (511 keV). With the detector field-of-view focused on the scatter line along Pv-Pd and the source collimated along the direction of Ps-Pv the electron density, xcfx81ev at Pv can be related to the detector response, S(E,xcex8) as follows:
S(E,xcex8)=k(E,xcex8)flxcfx81evfsxe2x80x83xe2x80x83(2)
where fl and fs are attenuation factors which account for the decrease in photon intensity as radiation travels toward and away from the scattering point, that is between the points Pin-Pv and between the points Pv-Pout respectivelly. k(E,xcex8) is a system constant that can be expressed, for a well-collimated source, as
k(E,xcex8)=S0D"sgr"(E)(p(cos xcex8)/2xcfx80R2)xcex7(Exe2x80x2)xe2x80x83xe2x80x83(3) 
where S0 is the source strength per unit area, D is voxel width, "sgr"(E) is the probability of scattering per unit area per electron (called microscopic scattering cross section) at energy E, p(cos xcex8) is the probability of a photon scattered at a specific angle xcex8, R is the distance from the scattering point to the detector, and xcex7(Exe2x80x2) is the detector efficiency at energy Exe2x80x2.
The incident and scattering attenuation factors (fl and fs) can be expressed as                                                                                           f                  i                                ⁢                                  xe2x80x83                                =                                  xe2x80x83                                ⁢                                  exp                  ⁡                                      [                                          -                                                                        ∫                          Pin                          Pr                                                ⁢                                                                                                                                            xe2x80x83                                                            ⁢                                                              μ                                t                                                                                      ⁢                                                          (                                                              r                                ,                                                                  xe2x80x83                                                                ⁢                                E                                                            )                                                                                ⁢                                                      xe2x80x83                                                    ⁢                                                      ⅆ                            r                                                                                                                ]                                                                                                                          =                                  exp                  [                                      xe2x80x83                                    ⁢                                      -                                                                  ∫                        Pin                        Pv                                            ⁢                                                                                                                                  xe2x80x83                                                        ⁢                                                          σ                              t                                                                                ⁢                                                      (                                                          r                              ,                                                              xe2x80x83                                                            ⁢                              E                                                        )                                                                          ⁢                                                  xe2x80x83                                                ⁢                                                                              ρ                            e                                                    (                                                      xe2x80x83                                                    ⁢                          r                          )                                                ⁢                                                  xe2x80x83                                                ⁢                                                  ⅆ                          r                                                                                                      ]                                                                    ⁢                  xe2x80x83                ⁢                  
                ⁢        and                            (        4        )                                                                                    f                s                            ⁢                              xe2x80x83                            =                              xe2x80x83                            ⁢                              exp                ⁡                                  [                                      -                                                                  ∫                        Pv                        Pout                                            ⁢                                                                                                                                  xe2x80x83                                                        ⁢                                                          μ                              t                                                                                ⁢                                                      (                                                                                          r                                xe2x80x2                                                            ,                                                              xe2x80x83                                                            ⁢                                                              E                                xe2x80x2                                                                                      )                                                                          ⁢                                                  xe2x80x83                                                ⁢                                                  ⅆ                                                      r                            xe2x80x2                                                                                                                                ]                                                                                                        =                              exp                [                                  xe2x80x83                                ⁢                                  -                                                            ∫                      Pv                      Pout                                        ⁢                                                                                                                        xe2x80x83                                                    ⁢                                                      σ                            t                                                                          ⁢                                                  (                                                                                    r                              xe2x80x2                                                        ,                                                          xe2x80x83                                                        ⁢                                                          E                              xe2x80x2                                                                                )                                                                    ⁢                                              xe2x80x83                                            ⁢                                                                        ρ                          e                                                (                                                  xe2x80x83                                                ⁢                                                  r                          xe2x80x2                                                )                                            ⁢                                              xe2x80x83                                            ⁢                                              ⅆ                                                  r                          xe2x80x2                                                                                                                    ]                                                                        (        5        )            
where xcexct(r,E) is the linear attenuation coefficient of photon of energy E at point (r), xcfx81e(r) is the electron density at point (r) along the beam path, and "sgr"t(r,E) is the total attenuation cross-section of photon of energy E per unit electron density by the material at point (r).
It will be appreciated that in order to calculate the electron density from the detector response, S(E,xcex8), as shown in equation (2), the attenuation factors, fl and fs must be determined. This has created problems in the past, since the values of these factors depend on the density of the material present in the path of the radiation beam, which are not usually known. Consequently, the formulation of the problem has been impracticable in the past because of the three unknowns in a single equation.
In the present invention, however, there is provided a method for determining with accuracy the density at a point within a region of interest in an object. This method is effected by generating and solving a set of fully-determined density-measurement equations.
Broadly, the method according to the present invention uses a X-ray beam having a relatively wide energy spectrum. Two source energy bands are selected within this source energy spectrum. Simultaneous measurements of the radiation transmitted and/or scattered are effected, at two distinct detected energy bands within the corresponding transmitted and/or scattered energy spectrums. The detected energy bands are calculated according to the unique relationship between the source energy bands and Compton scattering at a specific angle.
The source energy bands are selected in the energy range where Compton scattering dominates, and where the attenuation coefficient is directly proportional to density. This relation is used to reduce the attenuation factors at the two source energy bands to a single unknown as a function of energy and density.
Therefore, the main problem unknowns are; the attenuation factor along the incident beam and the density of the material of the object at the point within the region of interest. Each measurement adds with it one unknown, that is the attenuation factor along the path of the scattered or transmitted radiation. With the two main unknowns and each measurement adding with it one extra unknown, a system having N detectors yields 2+N unknowns. Keeping in mind that each detector provides two measurements, the number of measurements is 2N for N detectors. It will be appreciated that only two detectors are required to formulate a fully determined problem. A set of four equations, for example, using the radiation measurements of two detectors from the two source energy bands is sufficient to determine the attenuation of the radiation from the X-ray source to the point of interest, the attenuation of the radiation from the point of interest to each of the detectors and the density of the material of the object at the point of interest. Additional detectors or additional source energy bands can be used to over-determine the problem, and hence to increase the precision of the measurements.
This method is advantageous principally for allowing the measurement of the density at a point within an object without rotating the X-ray source, without rotating the radiation detectors and without rotating the object. The density measurement is effected without extrapolation from the measurements of neighboring voxels. The density measurement is effected with a single X-ray beam in a single exposure. The location of the examined voxel is determined by applying simple trigonometry principles.
It will be appreciated that density measurements can be obtain for a region of interest comprising several juxtaposed voxels within the object by moving the object between the X-ray source and the detectors. And of course, a three-dimensional density map of the entire object can be determined when needed, using point-by-point reconstruction of the object using well known computer-aided-drafting software.
In accordance with another aspect of the present invention, there is provided a best mode for simultaneously solving a set of fully-determined density-measurement equations. This best mode, or algorithm, consists of formulating the point density problem mentioned above into an invertible matrix that incorporates specific conditions, to increase the precision of determined density. The first condition requires that the matrix be a square matrix. Another condition requires that the source energy bands must be distinct from each other and far apart. A third condition requires that the detectors for measuring the scattered radiations be placed between the angles of 30xc2x0 and 165xc2x0 from the source X-ray beam, and preferably between 30xc2x0 and 60xc2x0 for the first detector and between 120xc2x0 and 150xc2x0 for the second detector. Finally, a third detector may be used to over-determine the problem, and this detector may be placed along the incident beam or at a right angle from it.
During experimental work, several sets of fully-determined density-measurement equations were successfully applied into an invertible matrix under the aforesaid conditions and simultaneously solved to determine the density of objects enclosed by other objects with relatively small differences with the actual values, that is below 5% difference.
Other advantages and novel features of the present invention will become apparent from the following detailed description.